Have you run across the Pythagorean Theorem yet? It's probably the most useful equation you'll take away from Geometry class - you'll use it everywhere. Fortunately it's pretty simple to remember.

A2 + B2 = C2

The Wikipedia article on the Pythagorean Theorem has a couple of good images to help show what this means, but in a nutshell:

Take a triangle. One that has a right angle. A right triangle.

"A" stands for the length of one side.

"B" stands for the length of another side.

"C" stands for the length of the third side - the one opposite the right angle.

Why am I talking about triangles when you only have two points? In your question, those two points are on a flat line - they have the same y-value (they're both seven units below the x-axis). But that may not be the case in your next problem. Fortunately, that doesn't keep us from solving your question here.

Let's make A represent the distance along the x-axis between your points (between 1 and 6). There are five units between those points (6 - 1 = 5, or 1 - 6= -5, either way it's five units forward or back from one to the other). So A is 5.

Let's have B represent the distance along the y-axis between your points (-7 and -7). They're both at the same height (have the same y-value), so there's no distance between them in that direction (-7 - -7 = -7 + 7 = 0). So B is zero.

Once you've got two of the three variables (A and B), you can find out what C is. Using the Pythagorean Theorem shown above (A2 + B2 = C2), fill in the values you know:

52 + 02 = C2

You can then solve the equation (see the steps below), but in this special case, you can also simplify it first if you know that 02 is 0, which adds nothing to the value and can be safely left out of the matter. Rewriting without the zero, you get

52 = C2

Take a look at it - if both sides of the equation are squared, then removing that exponent from both sides is ok, and leaves us with

5 = C

Which means that the length of the third leg of the triangle (the distance between those two points of yours) is 5.

The long way through that equation:

52 is 5 x 5 is 25

02 is 0 x 0 is 0

so

25 + 0 = C2

so

C = √25 (the square root of 25), which is 5.

This method works when the two points are not at the same level (have the same y-value) just as well.

Let's say that instead of points at (1,-7) and (6,-7), you have points at (10, 3) and (7, -1). Run those numbers through the Theorem....

.... no peeking....

Ok - here's what you probably got:

starting with

A2 + B2 = C2

you filled in what you know:

A = 10 - 7 = 3

B = 3 - -1 = 3 + 1 = 4

so

32 + 42 = C2

32 is 3 x 3, which is 9

42 is 4 x 4, which is 16

so

9 + 16 = C2

so

25 = C2

so
C = √25 (the square root of 25), which is 5.

No, the answer won't always be 5. It was here because I set it up that way - there's a special relationship between 3, 4, and 5 that is unique and shows up a LOT (32 + 42 = 52). Keep an eye out and you'll see that pattern everywhere - in buildings, in art, and on math tests.

For an answer that's not 5, try these points:

(1,1) and (1,3)

(6,-3) and (-2,-3)

(2,2) and (0,0) (hint: this last answer you'll have to just leave as the square root of something - which you'll do a lot in future classes)